The Properties And Judgment Of Generalized Diagonally Dominant Matrices

Generalized diagonally dominant matrices have a wide range of practical applications in biological engineering, control theory, differential equations, power system and other areas. Therefore, it has important theoretical significance and practical value to investigate the nature and determination of generalized diagonally dominant matrices. In this paper, some properties of generalized diagonally dominant matrices are obtained. Besides, Some sufficient and necessary conditions and sufficient conditions for a matrix to be a generalized diagonally dominant matrix or not to be a generalized diagonally dominant matrix are obtained by constructing several positive diagonal matrices and the comprehensive utilization of scaling the inequality techniques. Some numerical examples for the effectiveness and practicality of these criteria are presented. Partial results in this work develop and generalized the corresponding ones on diagonally dominant matrices, on nonsingular H-matrices. The paper is divided into five chapters, the structural framework is as follows:In chapter one, the background and significance are introduced and the main work are given.In chapter two, some basic symbols and definitions such as diagonally dominant matrices, generalized (strictly) diagonally dominant matrices, Hadamard product, Hadamard extract etc. are given.In chapter three, the properties of generalized diagonally dominant matrices are investigated. Conclusions such as the Hadamard product, Fan product. Hadamard extract (k>1) of two generalized diagonally dominant matrices is still a generalized diagonally dominant matrix are proved. The necessary condition for the Kronecker product of two matrices to be a generalized diagonally dominant matrix is also presented.In chapter four, the determination of generalized diagonally dominant matrices are investigated. Some sufficient and necessary conditions and sufficient conditions for a matrix to be a generalized diagonally dominant matrix or not to be a generalized diagonally dominant matrix are obtained by constructing several positive diagonal matrices and the comprehensive utilization of scaling the inequality techniques. The conclusion that a doubly diagonally dominant matrix is not necessarily a generalized diagonally dominant matrix is proved. Besides, a sufficient conditions for a double diagonally dominant matrix to be a generalized diagonally dominant matrix is presented.In chapter five, some numerical examples for the effectiveness and practicality of these criteria in chapter three and chapter four are given.